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Difference between revisions of "2012 AMC 10A Problems/Problem 24"

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== Problem 24 ==
 
== Problem 24 ==
  
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Let <math>a</math>, <math>b</math>, and <math>c</math> be positive integers with <math>a\geb\gec</math> such that
+
Let <math>a</math>, <math>b</math>, and <math>c</math> be positive integers with <math>a\ge</math> <math>b\ge</math> <math>c</math> such that
 
<math>a^2-b^2-c^2+ab=2011</math> and
 
<math>a^2-b^2-c^2+ab=2011</math> and
 
<math>a^2+3b^2+3c^2-3ab-2ac-2bc=-1997</math>.
 
<math>a^2+3b^2+3c^2-3ab-2ac-2bc=-1997</math>.

Revision as of 20:43, 8 February 2012

Problem 24

Let $a$, $b$, and $c$ be positive integers with $a\ge$ $b\ge$ $c$ such that $a^2-b^2-c^2+ab=2011$ and $a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$.

What is $a$?

$\textbf{(A)}\ 249\qquad\textbf{(B)}\ 250\qquad\textbf{(C)}\ 251\qquad\textbf{(D)}\ 252\qquad\textbf{(E)}\ 253$

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